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Suppose we have a ring homomorphism $\varphi: R \to S$, say an injection (e.g. coming from an injection $H \to G$ of finite groups and $R=\mathbb{Z}_p[H],S=\mathbb{Z}_p[G])$, what can be said about the kernel of $K_1(\varphi)$? Since I'm after all interested in Iwasawa-algebras let's suppose R,S are semilocal and by Vaserstein the canonical maps $i_R: R^\times\to K_1(R),~ i_S:S\times\to K_1(S)$ are surjective and we get have a comm. diagram $i_S\circ \varphi=K_1(\varphi)\circ i_R.$

There certainly are kernels sometimes: Let H be abelian, $G=H \rtimes C_2$ the semidirect produkt, where $C_2$ acts by inversion. Since $i_S$ factors through the abelianization of $S^{\times}$, we see that 2H is in the kernel.

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